10:00 – 12:00 Hermann Thorisson (University of Iceland)
Title: Coupling of Stochastic Processes.
Abstract: Coupling means the joint construction of two or more random variables, processes, or any random objects. The aim of the construction is usually to deduce properties of the individual objects or to gain insight into distributional relations between them. In these lectures we shall first consider some elementary examples, moving from Poisson approximation and stochastic domination to Markov chains and Brownian motion. We then outline a general coupling theory for stochastic processes and finally extend the view to random fields and to random elements under a topological transformation group. Applications to Markov processes, Brownian motion, regenerative processes and in Palm theory will be given along the way. The lectures will mainly be based on several chapters from a draft of the second edition of the book “Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, 2000”.
14:00 – 15:00 Wilfrid Kendall (University of Warwick)
Title: Probabilistic coupling and Nilpotent Diffusions.
Abstract: Modern probability theory makes considerable use of the technique of probabilistic coupling. The idea is, to analyse a given random process by constructing two inter-dependent copies of it, defined on the same probability space, and related in such a way as to facilitate analysis. Applications include: establishing monotonicity in non-obvious situations, developing quantitative approximations to distributions of random variables, constructing gradient estimates, and even producing exact simulation algorithms for Markov chains. However the thematic question, which has driven much of the theory of probabilistic coupling, concerns whether or not one can construct the two coupled random processes so that they almost surely meet (“couple”) at some future random time, and if so then whether one can construct a maximal coupling, for which the random time is smallest possible? The question is sharpened if we require the coupling to be co-adapted (also: immersed, or Markovian); this is an additional requirement that the coupling respect the underlying causal structure of the random processes, and can be viewed as implying that the coupling is easily constructable in some general sense. There is a considerable body of theory describing how to build successful co-adapted couplings for elliptic diffusions, all building on the basic reflection coupling for simple random walks or Brownian motion (very simply, the random jumps of the coupled process are arranged so far as possible to be the opposites of the random jumps of the original process). It is conjectured that successful co-adapted couplings can be built for all hypoelliptic diffusions (diffusions in d dimensions with fewer than d “directions of randomness”). In this talk I will survey the general theory of coupling, describe the known results for co-adapted couplings of hypoelliptic diffusions (in fact, Brownian motions on nilpotent Lie groups), and briefly discuss a related and very simple example which has applications to the theory of filtrations.
15:00 – 16:00 Hermann Thorisson (University of Iceland)
Title: Unbiased shifts of Brownian motion.
Abstract: Let $B=(B_t)_{tin R}$ be a two-sided standard Brownian motion. Let $T$ be a measurable function of $B$. Call $T$ an unbiased shift if $(B_{T+t}-B_T)_{tin R}$ is a Brownian motion independent of $B_T$. We characterize unbiased shifts in terms of allocation rules balancing local times of $B$. For any probability distribution $nu$ on $R$ we construct a nonnegative stopping time $T$ with the above properties such that $B_T$ has distribution $nu$. In particular, we show that if we travel in time according to the clock of local time at zero we always see a two-sided Brownian motion. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing jointly stationary diffuse random measures on $R$. We also study moment and minimality properties of unbiased shifts. The central results extend to recurrent Levy process under mild regularity conditions. This is based on joint work with G. Last and P. Moeters (to appear in the Annals of Probability).
16:00 – 17:00 Saul Jacka (University of Warwick)
Title: Coupling for pricing American options with stochastic volatility and model uncertainty.
Abstract: We discuss the pricing of American options under a stochastic volatility model and show how a coupling argument establishes that the exercise boundary for the option is monotone in the volatility. We will then go on to establish bounds for the price with (some) model uncertainty as to the stochastic volatility and discuss an extension to the case where we have a probabilistic version of model uncertainty.